A decomposition theorem for semiprime rings

A ring $A$ is called an $FDI$-ring if there exists a decomposition of the identity of $A$ in a sum of finite number of pairwise orthogonal primitive idempotents. We call a primitive idempotent $e$ artinian if the ring $eAe$ is Artinian. We prove that every semiprime $FDI$-ring is a direct product of a semisimple Artinian ring and a semiprime $FDI$-ring whose identity decomposition doesn't contain artinian idempotents.